One of the most popular topics in popular fitness media right now is "Zone 2 Training". This is a heart rate zone between 60-70% of your max (but it varies a bit from person to person).
In this zone, you're sweating, but you're not going so hard that you're really burning carbs for energy. You're mostly burning fat. What's more, you're creating the conditions to burn more fat in the future (maintaining lean muscle tissue, producing more mitochondria, and building fat-burning efficiency in other ways too.)
Here's a video on how it works.